Hi I am new to the site and I am uncertain how to do this question.
This is the questions, I am confused about. Click here for the image
So here is my working out/ideas about the question
a) I learned that differentiable is a function of one real variable is a function whose derivative exists at each point in its domain. However how do you apply that for $F$? So does that mean that that if $a = 0$ does that mean that it is differentiability .
b) Do I create a matrix for $F$, for instance
a b
c d
^ That is my attempt at conveying matrix
and multiply that by point $a$
c) Do I sub in (0,0) to show that it is continuous?
That is all I got so far, I know it isn't much but I am quite confused about this question.
Given a differentiable function $F: \mathbb{R}^n \to \mathbb{R}^m$, we have that the differential of $F$ is the linear map:
$$\\$$
$$d_pF = \left[\frac{\partial F^i}{\partial x^j}(p)\right]_{1 \leq i \leq m, i \leq j \leq n} \ \ \textbf{where}\ \ p \in U \subset \mathbb{R}^n$$
Example: If we let $f: \mathbb{R}^2 \to \mathbb{R}^3$ be defined by $f(x,y) =(F^1=x,F^2=y,F^3=x+y)$. Then we get that the differential is given by:
$$\\$$
$$d_pf = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}$$